Fluid_Mechanics_In_Si_Units_Fundamentals_and_Applications_4th_Edition.pdf

F L U I D M E C H A N I C S
FUNDAMENTALS AND APPLICATIONS
FOURTH EDITION IN SI UNITS
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A b o u t t h e A u t h o r s
Yunus A. Çengel is Professor Emeritus of Mechanical Engineering at the
University of Nevada, Reno. He received his B.S. in mechanical engineering from
Istanbul Technical University and his M.S. and Ph.D. in mechanical engineering
from North Carolina State University. His research areas are renewable energy,
desalination, exergy analysis, heat transfer enhancement, radiation heat transfer, and
energy conservation. He served as the director of the Industrial Assessment Center
(IAC) at the University of Nevada, Reno, from 1996 to 2000. He has led teams
of engineering students to numerous manufacturing facilities in Northern Nevada
and California to do industrial assessments, and has prepared energy conservation,
waste minimization, and productivity enhancement reports for them.
Dr. Çengel is the coauthor of the widely adopted textbook Thermodynamics: An Engi-
neering Approach, 8th edition (2015), published by McGraw-Hill Education. He is also
the coauthor of the textbook Heat and Mass Transfer: Fundamentals & Applications,
5th Edition (2015), and the coauthor of the textbook Fundamentals of Thermal-Fluid
Sciences, 5th edition (2017), both published by McGraw-Hill Education. Some of his
textbooks have been translated to Chinese, Japanese, Korean, Spanish, Turkish, Italian,
and Greek.
Dr. Çengel is the recipient of several outstanding teacher awards, and he has
received the ASEE Meriam/Wiley Distinguished Author Award for excellence in
authorship in 1992 and again in 2000.
Dr. Çengel is a registered Professional Engineer in the State of Nevada, and is a
member of the American Society of Mechanical Engineers (ASME) and the Ameri-
can Society for Engineering Education (ASEE).
John M. Cimbala is Professor of Mechanical Engineering at The Pennsyl-
vania State University, University Park. He received his B.S. in Aerospace Engi-
neering from Penn State and his M.S. in Aeronautics from the California Institute
of Technology (CalTech). He received his Ph.D. in Aeronautics from CalTech in
1984 under the supervision of Professor Anatol Roshko, to whom he will be forever
grateful. His research areas include experimental and computational fluid mechan-
ics and heat transfer, turbulence, turbulence modeling, turbomachinery, indoor air
quality, and air pollution control. Professor Cimbala completed sabbatical leaves
at NASA Langley Research Center (1993–94), where he advanced his knowledge
of computational fluid dynamics (CFD), and at Weir American Hydo (2010–11),
where he performed CFD analyses to assist in the design of hydroturbines.
Dr. Cimbala is the coauthor of three other textbooks: Indoor Air Quality Engi-
neering: Environmental Health and Control of Indoor Pollutants (2003), pub-
lished by Marcel-Dekker, Inc.; Essentials of Fluid Mechanics: Fundamentals and
Applications (2008); and Fundamentals of Thermal-Fluid Sciences, 5th edition
(2017), both published by McGraw-Hill Education. He has also contributed to parts
of other books, and is the author or coauthor of dozens of journal and conference
papers. He has also recently ventured into writing novels. More information can be
found at www.mne.psu.edu/cimbala.
Professor Cimbala is the recipient of several outstanding teaching awards and
views his book writing as an extension of his love of teaching. He is a member of
the American Society of Mechanical Engineers (ASME), the American Society for
Engineering Education (ASEE), and the American Physical Society (APS).

F L U I D M E C H A N I C S
FUNDAMENTALS AND APPLICATIONS
YUNUS A.
ÇENGEL
Department of
Mechanical
Engineering
University of Nevada,
Reno
JOHN M.
CIMBALA
Department of
Mechanical and
Nuclear Engineering
The Pennsylvania
State University
Adapted by
MEHMET KANOĞLU
University of Gaziantep

FLUID MECHANICS: FUNDAMENTALS AND APPLICATIONS
Copyright ©2020 by McGraw-Hill Education. All rights reserved. Previous editions © 2014, 2010, and 2006. No part of this publication may
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B r i e f C o n t e n t s
c h a p t e r o n e
INTRODUCTION AND BASIC CONCEPTS 1
c h a p t e r t w o
PROPERTIES OF FLUIDS 37
c h a p t e r t h r e e
PRESSURE AND FLUID STATICS 77
c h a p t e r f o u r
FLUID KINEMATICS 137
c h a p t e r f i v e
BERNOULLI AND ENERGY EQUATIONS 189
c h a p t e r s i x
MOMENTUM ANALYSIS OF FLOW SYSTEMS 249
c h a p t e r s e v e n
DIMENSIONAL ANALYSIS AND MODELING 297
c h a p t e r e i g h t
INTERNAL FLOW 351
c h a p t e r n i n e
DIFFERENTIAL ANALYSIS OF FLUID FLOW 443
c h a p t e r t e n
APPROXIMATE SOLUTIONS OF THE NAVIER–STOKES
EQUATION 519
c h a p t e r e l e v e n
EXTERNAL FLOW: DRAG AND LIFT 611
c h a p t e r t w e l v e
COMPRESSIBLE FLOW 667
c h a p t e r t h i r t e e n
OPEN-CHANNEL FLOW 733
c h a p t e r f o u r t e e n
TURBOMACHINERY 793
c h a p t e r f i f t e e n
INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS 885

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Preface xv
c h a p t e r o n e
INTRODUCTION AND BASIC
CONCEPTS 1
1–1 Introduction 2
What Is a Fluid? 2
Application Areas of Fluid Mechanics 4
1–2 A Brief History of Fluid Mechanics 6
1–3 The No-Slip Condition 8
1–4 Classification of Fluid Flows 9
Viscous versus Inviscid Regions of Flow 10
Internal versus External Flow 10
Compressible versus Incompressible Flow 10
Laminar versus Turbulent Flow 11
Natural (or Unforced) versus Forced Flow 11
Steady versus Unsteady Flow 12
One-, Two-, and Three-Dimensional Flows 13
Uniform versus Nonuniform Flow 14
1–5 System and Control Volume 15
1–6 Importance of Dimensions and Units 16
Some SI and English Units 17
Dimensional Homogeneity 19
Unity Conversion Ratios 21
1–7
Modeling in Engineering 22
1–8 Problem-Solving Technique 24
Step 1: Problem Statement 24
Step 2: Schematic 24
Step 3: Assumptions and Approximations 24
Step 4: Physical Laws 24
Step 5: Properties 25
Step 6: Calculations 25
Step 7: Reasoning, Verification, and Discussion 25
1–9
Engineering Software Packages 26
Equation Solvers 27
CFD Software 28
1–10
Accuracy, Precision, and Significant Digits 28
Application Spotlight: What Nuclear Blasts and
Raindrops Have in Common 32
Summary 33
References and Suggested Reading 33
Problems 33
c h a p t e r t w o
PROPERTIES OF FLUIDS 37
Continuum 38
2–2
Density and Specific Gravity 39
Density of Ideal Gases 40
2–3
Vapor Pressure and Cavitation 41
2–4
Energy and Specific Heats 43
2–5
Compressibility and Speed of Sound 45
Coefficient of Compressibility 45
Coefficient of Volume Expansion 46
Speed of Sound and Mach Number 49
2–6 Viscosity 51
2–7
Surface Tension and Capillary Effect 56
Capillary Effect 59
Summary 62
Application Spotlight: Cavitation 63
Problems 64
c h a p t e r t h r e e
PRESSURE AND FLUID STATICS 77
3–1 Pressure 78
Pressure at a Point 79
Variation of Pressure with Depth 80
3–2
Pressure Measurement Devices 84
The Barometer 84
The Manometer 87
Other Pressure Measurement Devices 90
3–3
Introduction to Fluid Statics 91
3–4
Hydrostatic Forces on Submerged
Plane Surfaces 92
Special Case: Submerged Rectangular Plate 95
3–5
Hydrostatic Forces on Submerged Curved
Surfaces 97
3–6
Buoyancy and Stability 100
Stability of Immersed and Floating Bodies 104
C o n t e n t s

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FLUID MECHANICS
x
CONTENTS
3–7
Fluids in Rigid-Body Motion 106
Special Case 1: Fluids at Rest 108
Special Case 2: Free Fall of a Fluid Body 108
Acceleration on a Straight Path 108
Rotation in a Cylindrical Container 110
Summary 114
Problems 115
c h a p t e r f o u r
FLUID KINEMATICS 137
4–1
Lagrangian and Eulerian
Descriptions 138
Acceleration Field 140
Material Derivative 143
4–2 Flow Patterns and Flow Visualization 145
Streamlines and Streamtubes 145
Pathlines 146
Streaklines 148
Timelines 150
Refractive Flow Visualization Techniques 151
Surface Flow Visualization Techniques 152
4–3 Plots of Fluid Flow Data 152
Profile Plots 153
Vector Plots 153
Contour Plots 154
4–4 Other Kinematic Descriptions 155
Types of Motion or Deformation of Fluid
Elements 155
4–5 Vorticity and Rotationality 160
Comparison of Two Circular Flows 163
4–6 The Reynolds Transport Theorem 164
Alternate Derivation of the Reynolds Transport
Theorem 169
Relationship between Material Derivative and RTT 172
Summary 172
Application Spotlight: Fluidic Actuators 173
Application Spotlight: Smelling Food; the
Human Airway 174
Problems 175
c h a p t e r f i v e
BERNOULLI AND ENERGY
EQUATIONS 189
Conservation of Mass 190
The Linear Momentum Equation 190
Conservation of Energy 190
5–2 Conservation of Mass 191
Mass and Volume Flow Rates 191
Conservation of Mass Principle 193
Moving or Deforming Control Volumes 195
Mass Balance for Steady-Flow Processes 195
Special Case: Incompressible Flow 196
5–3 Mechanical Energy and Efficiency 198
5–4 The Bernoulli Equation 203
Acceleration of a Fluid Particle 204
Derivation of the Bernoulli Equation 204
Force Balance across Streamlines 206
Unsteady, Compressible Flow 207
Static, Dynamic, and Stagnation Pressures 207
Limitations on the Use of the Bernoulli Equation 208
Hydraulic Grade Line (HGL) and Energy
Grade Line (EGL) 210
Applications of the Bernoulli Equation 212
5–5 General Energy Equation 219
Energy Transfer by Heat, Q 220
Energy Transfer by Work, W 220
5–6 Energy Analysis of Steady Flows 223
Special Case: Incompressible Flow with No Mechanical
Work Devices and Negligible Friction 226
Kinetic Energy Correction Factor, 𝛼 226
Summary 233
Problems 235
c h a p t e r s i x
MOMENTUM ANALYSIS OF FLOW
SYSTEMS 249
6–1 Newton’s Laws 250
6–2 Choosing a Control Volume 251
6–3 Forces Acting on a Control Volume 252
6–4 The Linear Momentum Equation 255
Special Cases 257
Momentum-Flux Correction Factor, β 257
Steady Flow 259
Flow with No External Forces 260
6–5 Review of Rotational Motion and Angular
Momentum 269
6–6
The Angular Momentum Equation 272
Special Cases 274
Flow with No External Moments 275
Radial-Flow Devices 275

CONTENTS
xi
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Application Spotlight: Manta Ray
Swimming 280
Summary 282
Problems 283
c h a p t e r s e v e n
DIMENSIONAL ANALYSIS AND
MODELING 297
7–1 Dimensions and Units 298
7–2 Dimensional Homogeneity 299
Nondimensionalization of Equations 300
7–3 Dimensional Analysis and Similarity 305
7–4 The Method of Repeating Variables and the
Buckingham Pi Theorem 309
Historical Spotlight: Persons Honored by
Nondimensional Parameters 317
7–5
Experimental Testing, Modeling, and Incomplete
Similarity 325
Setup of an Experiment and Correlation
of Experimental Data 325
Incomplete Similarity 326
Wind Tunnel Testing 326
Flows with Free Surfaces 329
Application Spotlight: How a Fly Flies 332
Summary 333
Problems 333
c h a p t e r e i g h t
INTERNAL FLOW 351
8–2 Laminar and Turbulent Flows 353
Reynolds Number 354
8–3 The Entrance Region 355
Entry Lengths 356
8–4 Laminar Flow in Pipes 357
Pressure Drop and Head Loss 359
Effect of Gravity on Velocity and Flow Rate
in Laminar Flow 361
Laminar Flow in Noncircular Pipes 362
8–5 Turbulent Flow in Pipes 365
Turbulent Shear Stress 366
Turbulent Velocity Profile 368
The Moody Chart and Its Associated
Equations 370
Types of Fluid Flow Problems 372
8–6 Minor Losses 379
8–7 Piping Networks and Pump
Selection 386
Series and Parallel Pipes 386
Piping Systems with Pumps and Turbines 388
8–8 Flow Rate and Velocity
Measurement 396
Pitot and Pitot-Static Probes 396
Obstruction Flowmeters: Orifice, Venturi,
and Nozzle Meters 398
Positive Displacement Flowmeters 401
Turbine Flowmeters 402
Variable-Area Flowmeters (Rotameters) 403
Ultrasonic Flowmeters 404
Electromagnetic Flowmeters 406
Vortex Flowmeters 407
Thermal (Hot-Wire and Hot-Film)
Anemometers 408
Laser Doppler Velocimetry 410
Particle Image Velocimetry 411
Introduction to Biofluid Mechanics 414
Application Spotlight: PIV Applied to Cardiac
Flow 420
Application Spotlight: Multicolor Particle
Shadow Velocimetry/Accelerometry 421
Summary 423
Problems 425
c h a p t e r n i n e
DIFFERENTIAL ANALYSIS OF FLUID
FLOW 443
9–2 Conservation of Mass—The Continuity
Equation 444
Derivation Using the Divergence Theorem 445
Derivation Using an Infinitesimal Control
Volume 446
Alternative Form of the Continuity Equation 449
Continuity Equation in Cylindrical Coordinates 450
Special Cases of the Continuity Equation 450
9–3 The Stream Function 456
The Stream Function in Cartesian Coordinates 456
The Stream Function in Cylindrical Coordinates 463
The Compressible Stream Function 464

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CONTENTS
9–4 The Differential Linear Momentum Equation—
Cauchy’s Equation 465
Derivation Using the Divergence Theorem 465
Derivation Using an Infinitesimal Control
Volume 466
Alternative Form of Cauchy’s Equation 469
Derivation Using Newton’s Second Law 469
9–5 The Navier–Stokes Equation 470
Introduction 470
Newtonian versus Non-Newtonian Fluids 471
Derivation of the Navier–Stokes Equation
for Incompressible, Isothermal Flow 472
Continuity and Navier–Stokes Equations
in Cartesian Coordinates 474
Continuity and Navier–Stokes Equations
in Cylindrical Coordinates 475
9–6 Differential Analysis of Fluid Flow
Problems 476
Calculation of the Pressure Field
for a Known Velocity Field 476
Exact Solutions of the Continuity
and Navier–Stokes Equations 481
Differential Analysis of Biofluid Mechanics Flows 499
Summary 502
Application Spotlight: The No-Slip Boundary
Condition 503
Problems 504
c h a p t e r t e n
APPROXIMATE SOLUTIONS OF THE
NAVIER–STOKES EQUATION 519
10–2
Nondimensionalized Equations of Motion 521
10–3
The Creeping Flow Approximation 524
Drag on a Sphere in Creeping Flow 527
10–4 Approximation for Inviscid Regions of Flow 529
Derivation of the Bernoulli Equation in Inviscid Regions
of Flow 530
10–5 The Irrotational Flow
Approximation 533
Continuity Equation 533
Momentum Equation 535
Derivation of the Bernoulli Equation in Irrotational
Regions of Flow 535
Two-Dimensional Irrotational Regions of Flow 538
Superposition in Irrotational Regions of Flow 542
Elementary Planar Irrotational Flows 542
Irrotational Flows Formed by Superposition 549
10–6 The Boundary Layer
Approximation 558
The Boundary Layer Equations 563
The Boundary Layer Procedure 568
Displacement Thickness 572
Momentum Thickness 575
Turbulent Flat Plate Boundary Layer 576
Boundary Layers with Pressure Gradients 582
The Momentum Integral Technique for Boundary Layers 587
Summary 595
Application Spotlight: Droplet Formation 597
Problems 598
c h a p t e r e l e v e n
EXTERNAL FLOW: DRAG AND LIFT 611
11–2 Drag and Lift 614
11–3 Friction and Pressure Drag 618
Reducing Drag by Streamlining 619
Flow Separation 620
11–4 Drag Coefficients of Common Geometries 621
Biological Systems and Drag 622
Drag Coefficients of Vehicles 625
Superposition 627
11–5 Parallel Flow over Flat Plates 629
Friction Coefficient 631
11–6 Flow over Cylinders and Spheres 633
Effect of Surface Roughness 636
11–7 Lift 638
Finite-Span Wings and Induced Drag 642
Lift Generated by Spinning 643
Flying in Nature! 647
Summary 650
Application Spotlight: Drag Reduction 652
Problems 653
c h a p t e r t w e l v e
COMPRESSIBLE FLOW 667
12–1 Stagnation Properties 668
12–2 One-Dimensional Isentropic Flow 671
Variation of Fluid Velocity with Flow Area 673
Property Relations for Isentropic Flow of Ideal Gases 675

CONTENTS
xiii
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12–3 Isentropic Flow through Nozzles 677
Converging Nozzles 678
Converging–Diverging Nozzles 682
12–4 Shock Waves and Expansion
Waves 685
Normal Shocks 686
Oblique Shocks 691
Prandtl–Meyer Expansion Waves 696
12–5 Duct Flow with Heat Transfer and Negligible
Friction (Rayleigh Flow) 701
Property Relations for Rayleigh Flow 706
Choked Rayleigh Flow 708
12–6
Adiabatic Duct Flow with Friction
(Fanno Flow) 710
Property Relations for Fanno Flow 713
Choked Fanno Flow 716
Application Spotlight: Shock-Wave/
Boundary-Layer Interactions 720
Summary 721
Problems 722
c h a p t e r t h i r t e e n
OPEN-CHANNEL FLOW 733
13–1
Classification of Open-Channel Flows 734
Uniform and Varied Flows 734
Laminar and Turbulent Flows in Channels 735
13–2 Froude Number and Wave Speed 737
Speed of Surface Waves 739
13–3 Specific Energy 741
13–4 Conservation of Mass and Energy
Equations 744
13–5 Uniform Flow in Channels 745
Critical Uniform Flow 747
Superposition Method for Nonuniform
Perimeters 748
13–6 Best Hydraulic Cross Sections 751
Rectangular Channels 753
Trapezoidal Channels 753
13–7 Gradually Varied Flow 755
Liquid Surface Profiles in Open Channels, y(x) 757
Some Representative Surface Profiles 760
Numerical Solution of Surface Profile 762
13–8 Rapidly Varied Flow and the Hydraulic
Jump 765
13–9 Flow Control and Measurement 769
Underflow Gates 770
Overflow Gates 772
Application Spotlight: Bridge Scour 779
Summary 780
Problems 781
c h a p t e r f o u r t e e n
TURBOMACHINERY 793
14–1 Classifications and Terminology 794
14–2 Pumps 796
Pump Performance Curves and Matching
a Pump to a Piping System 797
Pump Cavitation and Net Positive Suction Head 803
Pumps in Series and Parallel 806
Positive-Displacement Pumps 809
Dynamic Pumps 812
Centrifugal Pumps 812
Axial Pumps 822
14–3 Pump Scaling Laws 830
Dimensional Analysis 830
Pump Specific Speed 833
Affinity Laws 835
14–4 Turbines 839
Positive-Displacement Turbines 841
Dynamic Turbines 841
Impulse Turbines 842
Reaction Turbines 843
Gas and Steam Turbines 853
Wind Turbines 853
14–5 Turbine Scaling Laws 861
Dimensionless Turbine Parameters 861
Turbine Specific Speed 864
Application Spotlight: Rotary Fuel Atomizers 867
Summary 868
Problems 869
c h a p t e r f i f t e e n
INTRODUCTION TO COMPUTATIONAL
FLUID DYNAMICS 885
15–1 Introduction and Fundamentals 886
Motivation 886
Equations of Motion 886
Solution Procedure 887
Additional Equations of Motion 889

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CONTENTS
Grid Generation and Grid Independence 889
Boundary Conditions 894
Practice Makes Perfect 899
15–2 Laminar CFD Calculations 899
Pipe Flow Entrance Region at Re = 500 899
Flow around a Circular Cylinder at Re = 150 903
15–3 Turbulent CFD Calculations 908
Flow around a Circular Cylinder at Re = 10,000 911
Flow around a Circular Cylinder at Re = 107 913
Design of the Stator for a Vane-Axial Flow Fan 913
15–4 CFD with Heat Transfer 921
Temperature Rise through a Cross-Flow Heat
Exchanger 921
Cooling of an Array of Integrated Circuit Chips 923
15–5 Compressible Flow CFD Calculations 928
Compressible Flow through a Converging–Diverging
Nozzle 929
Oblique Shocks over a Wedge 933
CFD Methods for Two-Phase Flows 934
15–6 Open-Channel Flow CFD Calculations 936
Flow over a Bump on the Bottom of a Channel 936
Flow through a Sluice Gate (Hydraulic Jump) 937
Summary 938
Application Spotlight: A Virtual Stomach 939
Problems 940
a p p e n d i x
PROPERTY TABLES AND CHARTS 947
TABLE A–1 Molar Mass, Gas Constant, and
Ideal-Gas Specific Heats of Some
Substances 948
TABLE A–2 Boiling and Freezing Point
Properties 949
TABLE A–3 Properties of Saturated Water 950
TABLE A–4 Properties of Saturated
Refrigerant-134a 951
TABLE A–5 Properties of Saturated Ammonia 952
TABLE A–6 Properties of Saturated Propane 953
TABLE A–7 Properties of Liquids 954
TABLE A–8 Properties of Liquid Metals 955
TABLE A–9 Properties of Air at 1 atm Pressure 956
TABLE A–10 Properties of Gases at 1 atm
Pressure 957
TABLE A–11 Properties of the Atmosphere at High
Altitude 959
FIGURE A–12 The Moody Chart for the Friction
Factor for Fully Developed Flow in
Circular Pipes 960
TABLE A–13 One-Dimensional Isentropic
Compressible Flow Functions for an
Ideal Gas with k = 1.4 961
TABLE A–14 One-Dimensional Normal Shock
Functions for an Ideal Gas with
k = 1.4 962
TABLE A–15 Rayleigh Flow Functions for an Ideal
Gas with k = 1.4 963
TABLE A–16 Fanno Flow Functions for an Ideal Gas
with k = 1.4 964
Glossary 965
Index 979
Conversion Factors 995
Nomenclature 997

P r e f a c e
BACKGROUND
Fluid mechanics is an exciting and fascinating subject with unlimited practi-
cal applications ranging from microscopic biological systems to automobiles,
airplanes, and spacecraft propulsion. Fluid mechanics has also historically
been one of the most challenging subjects for undergraduate students because
proper analysis of fluid mechanics problems requires not only knowledge
of the concepts but also physical intuition and experience. Our hope is that
this book, through its careful explanations of concepts and its use of numer-
ous practical examples, sketches, figures, and photographs, bridges the gap
between knowledge and the proper application of that knowledge.
Fluid mechanics is a mature subject; the basic equations and approxima-
tions are well established and can be found in any introductory textbook. Our
book is distinguished from other introductory books because we present the
subject in a progressive order from simple to more difficult, building each
chapter upon foundations laid down in earlier chapters. We provide more dia-
grams and photographs than other books because fluid mechanics is, by its
nature, a highly visual subject. Only by illustrating the concepts discussed,
can students fully appreciate the mathematical significance of the material.
OBJECTIVES
This book has been written for the first fluid mechanics course for under-
graduate engineering students. There is sufficient material for a two-course
sequence, if desired. We assume that readers will have an adequate back-
ground in calculus, physics, engineering mechanics, and thermodynamics.
The objectives of this text are
⬤
⬤
To present the basic principles and equations of fluid mechanics.
⬤
⬤
To show numerous and diverse real-world engineering examples to
give the student the intuition necessary for correct application of fluid
mechanics principles in engineering applications.
⬤
⬤
To develop an intuitive understanding of fluid mechanics by emphasiz-
ing the physics, and reinforcing that understanding through illustrative
figures and photographs.
The book contains enough material to allow considerable flexibility in teach-
ing the course. Aeronautics and aerospace engineers might emphasize poten-
tial flow, drag and lift, compressible flow, turbomachinery, and CFD, while
mechanical or civil engineering instructors might choose to emphasize pipe
flows and open-channel flows, respectively.
NEW TO THE FOURTH EDITION
All the popular features of the previous editions have been retained while new
ones have been added. The main body of the text remains largely unchanged.
A noticeable change is the addition of a number of exciting new pictures
throughout the book.

xvi
PREFACE
Four new subsections have been added: “Uniform versus Nonuniform
Flow” and “Equation Solvers” to Chap. 1, “Flying in Nature” by guest author
Azar Eslam Panah of Penn State Berks to Chap. 11, and “CFD Methods for
Two-Phase Flows” by guest author Alex Rattner of Penn State to Chap. 15. In
Chap. 8, we now highlight the explicit Churchill equation as an alternative to
the implicit Colebrook equation.
Two new Application Spotlights, have been added: “Smelling Food; the
Human Airway” by Rui Ni of Penn State, to Chap. 4, and “Multicolor Par-
ticle Shadow Velocimetry/Accelerometry” by Michael McPhail and Michael
Krane of Penn State to Chap. 8.
A large number of the end-of-chapter problems in the text have been mod-
ified and many problems were replaced by new ones. Also, several of the
solved example problems have been replaced.
PHILOSOPHY AND GOAL
The Fourth Edition of Fluid Mechanics: Fundamentals and Applications has
the same goals and philosophy as the other texts by lead author Yunus Çengel.
⬤
⬤
Communicates directly with tomorrow’s engineers in a simple yet
precise manner
⬤
⬤
Leads students toward a clear understanding and firm grasp of the basic
principles of fluid mechanics
⬤
⬤
Encourages creative thinking and development of a deeper understand-
ing and intuitive feel for fluid mechanics
⬤
⬤
Is read by students with interest and enthusiasm rather than merely as a
guide to solve homework problems
The best way to learn is by practice. Special effort is made throughout the
book to reinforce the material that was presented earlier (in each chapter
as well as in material from previous chapters). Many of the illustrated
example problems and end-of-chapter problems are comprehensive and
encourage students to review and revisit concepts and intuitions gained
previously.
Throughout the book, we show examples generated by computational fluid
dynamics (CFD). We also provide an introductory chapter on the subject. Our
goal is not to teach the details about numerical algorithms associated with
CFD—this is more properly presented in a separate course. Rather, our intent
is to introduce undergraduate students to the capabilities and limitations of
CFD as an engineering tool. We use CFD solutions in much the same way
as experimental results are used from wind tunnel tests (i.e., to reinforce
understanding of the physics of fluid flows and to provide quality flow visual-
izations that help explain fluid behavior). With dozens of CFD end-of-chapter
problems posted on the website, instructors have ample opportunity to intro-
duce the basics of CFD throughout the course.
CONTENT AND ORGANIZATION
This book is organized into 15 chapters beginning with fundamental concepts
of fluids, fluid properties, and fluid flows and ending with an introduction to
computational fluid dynamics.
⬤
⬤
Chapter 1 provides a basic introduction to fluids, classifications of fluid
flow, control volume versus system formulations, dimensions, units,
significant digits, and problem-solving techniques.

⬤
⬤
Chapter 2 is devoted to fluid properties such as density, vapor pressure,
specific heats, speed of sound, viscosity, and surface tension.
⬤
⬤
Chapter 3 deals with fluid statics and pressure, including manometers
and barometers, hydrostatic forces on submerged surfaces, buoyancy
and stability, and fluids in rigid-body motion.
⬤
⬤
Chapter 4 covers topics related to fluid kinematics, such as the differ-
ences between Lagrangian and Eulerian descriptions of fluid flows,
flow patterns, flow visualization, vorticity and rotationality, and the
Reynolds transport theorem.
⬤
⬤
Chapter 5 introduces the fundamental conservation laws of mass,
momentum, and energy, with emphasis on the proper use of the mass,
Bernoulli, and energy equations and the engineering applications of
these equations.
⬤
⬤
Chapter 6 applies the Reynolds transport theorem to linear momentum
and angular momentum and emphasizes practical engineering applica-
tions of finite control volume momentum analysis.
⬤
⬤
Chapter 7 reinforces the concept of dimensional homogeneity and intro-
duces the Buckingham Pi theorem of dimensional analysis, dynamic
similarity, and the method of repeating variables—material that is use-
ful throughout the rest of the book and in many disciplines in science
and engineering.
⬤
⬤
Chapter 8 is devoted to flow in pipes and ducts. We discuss the dif-
ferences between laminar and turbulent flow, friction losses in pipes
and ducts, and minor losses in piping networks. We also explain how
to properly select a pump or fan to match a piping network. Finally, we
discuss various experimental devices that are used to measure flow rate
and velocity, and provide a brief introduction to biofluid mechanics.
⬤
⬤
Chapter 9 deals with differential analysis of fluid flow and includes der-
ivation and application of the continuity equation, the Cauchy equation,
and the Navier–Stokes equation. We also introduce the stream function
and describe its usefulness in analysis of fluid flows, and we provide a
brief introduction to biofluids. Finally, we point out some of the unique
aspects of differential analysis related to biofluid mechanics.
⬤
⬤
Chapter 10 discusses several approximations of the Navier–Stokes equa-
tion and provides example solutions for each approximation, including
creeping flow, inviscid flow, irrotational (potential) flow, and boundary
layers.
⬤
⬤
Chapter 11 covers forces on living and non-living bodies (drag and
lift), explaining the distinction between friction and pressure drag,
and providing drag coefficients for many common geometries. This
chapter emphasizes the practical application of wind tunnel mea-
surements coupled with dynamic similarity and dimensional analysis
concepts introduced earlier in Chap. 7.
⬤
⬤
Chapter 12 extends fluid flow analysis to compressible flow, where the
behavior of gases is greatly affected by the Mach number. In this chapter,
the concepts of expansion waves, normal and oblique shock waves, and
choked flow are introduced.
⬤
⬤
Chapter 13 deals with open-channel flow and some of the unique fea-
tures associated with the flow of liquids with a free surface, such as
surface waves and hydraulic jumps.
PREFACE
xvii

xviii
PREFACE
⬤
⬤
Chapter 14 examines turbomachinery in more detail, including pumps,
fans, and turbines. An emphasis is placed on how pumps and turbines
work, rather than on their detailed design. We also discuss overall pump
and turbine design, based on dynamic similarity laws and simplified
velocity vector analyses.
⬤
⬤
Chapter 15 describes the fundamental concepts of computational fluid
dyamics (CFD) and shows students how to use commercial CFD codes
as tools to solve complex fluid mechanics problems. We emphasize the
application of CFD rather than the algorithms used in CFD codes.
Each chapter contains a wealth of end-of-chapter homework problems. A
comprehensive set of appendices is provided, giving the thermodynamic and
fluid properties of several materials, in addition to air and water, along with
some useful plots and tables. Many of the end-of-chapter problems require
the use of material properties from the appendices to enhance the realism of
the problems.
LEARNING TOOLS
EMPHASIS ON PHYSICS
A distinctive feature of this book is its emphasis on the physical aspects
of the subject matter in addition to mathematical representations and
manipulations. The authors believe that the emphasis in undergraduate
education should remain on developing a sense of underlying physical
mechanisms and a mastery of solving practical problems that an engineer
is likely to face in the real world. Developing an intuitive understanding
should also make the course a more motivating and worthwhile experi-
ence for the students.
EFFECTIVE USE OF ASSOCIATION
An observant mind should have no difficulty understanding engineering
sciences. After all, the principles of engineering sciences are based on our
everyday experiences and experimental observations. Therefore, a physi-
cal, intuitive approach is used throughout this text. Frequently, parallels are
drawn between the subject matter and students’ everyday experiences so that
they can relate the subject matter to what they already know.
SELF-INSTRUCTING
The material in the text is introduced at a level that an average student can
follow comfortably. It speaks to students, not over students. In fact, it is self-
instructive. Noting that the principles of science are based on experimental
observations, most of the derivations in this text are largely based on physical
arguments, and thus they are easy to follow and understand.
EXTENSIVE USE OF ARTWORK AND PHOTOGRAPHS
Figures are important learning tools that help the students “get the picture,”
and the text makes effective use of graphics. It contains more figures, photo-
graphs, and illustrations than any other book in this category. Figures attract
attention and stimulate curiosity and interest. Most of the figures in this text

are intended to serve as a means of emphasizing some key concepts that
would otherwise go unnoticed; some serve as page summaries.
NUMEROUS WORKED-OUT EXAMPLES
All chapters contain numerous worked-out examples that both clarify the
material and illustrate the use of basic principles in a context that helps
develop the student’s intuition. An intuitive and systematic approach is used
in the solution of all example problems. The solution methodology starts with
a statement of the problem, and all objectives are identified. The assumptions
and approximations are then stated together with their justifications. Any
properties needed to solve the problem are listed separately. Numerical values
are used together with numbers to emphasize that without units, numbers are
meaningless. The significance of each example’s result is discussed following
the solution. This methodical approach is also followed and provided in the
solutions to the end-of-chapter problems, available to instructors.
A WEALTH OF REALISTIC END-OF-CHAPTER PROBLEMS
The end-of-chapter problems are grouped under specific topics to make
problem selection easier for both instructors and students. Within each
group of problems are Concept Questions, indicated by “C,” to check the
students’ level of understanding of basic concepts. Problems under Funda-
mentals of Engineering (FE) Exam Problems are designed to help students
prepare for the Fundamentals of Engineering exam, as they prepare
for their Professional Engineering license. The problems under Review
Problems are more comprehensive in nature and are not directly tied
to any specific section of a chapter—in some cases they require review
of material learned in previous chapters. Problems designated as
Design and Essay are intended to encourage students to make engineering
judgments, to conduct independent exploration of topics of interest, and to
communicate their findings in a professional manner. Problems with the
icon are comprehensive in nature and are intended to be solved with a
computer, using appropriate software. Several economics- and safety-related
problems are incorporated throughout to enhance cost and safety awareness
among engineering students. Answers to selected problems are listed imme-
diately following the problem for convenience to students.
USE OF COMMON NOTATION
The use of different notation for the same quantities in different engineering
courses has long been a source of discontent and confusion. A student taking
both fluid mechanics and heat transfer, for example, has to use the notation Q
for volume flow rate in one course, and for heat transfer in the other. The need
to unify notation in engineering education has often been raised, even in some
reports of conferences sponsored by the National Science Foundation through
Foundation Coalitions, but little effort has been made to date in this regard.
For example, refer to the final report of the Mini-Conference on Energy Stem
Innovations, May 28 and 29, 2003, University of Wisconsin. In this text we
made a conscious effort to minimize this conflict by adopting the familiar
thermodynamic notation V̇ for volume flow rate, thus reserving the notation Q
PREFACE
xix

for heat transfer. Also, we consistently use an overdot to denote time rate. We
think that both students and instructors will appreciate this effort to promote
a common notation.
COMBINED COVERAGE OF BERNOULLI
AND ENERGY EQUATIONS
The Bernoulli equation is one of the most frequently used equations in fluid
mechanics, but it is also one of the most misused. Therefore, it is important
to emphasize the limitations on the use of this idealized equation and to
show how to properly account for imperfections and irreversible losses.
In Chap. 5, we do this by introducing the energy equation right after the

Bernoulli equation and demonstrating how the solutions of many practical
engineering problems differ from those obtained using the Bernoulli equa-
tion. This helps students develop a realistic view of the Bernoulli equation.
A SEPARATE CHAPTER ON CFD
Commercial Computational Fluid Dynamics (CFD) codes are widely used
in engineering practice in the design and analysis of flow systems, and it has
become exceedingly important for engineers to have a solid understanding of
the fundamental aspects, capabilities, and limitations of CFD. Recognizing
that most undergraduate engineering curriculums do not have room for a full
course on CFD, a separate chapter is included here to make up for this defi-
ciency and to equip students with an adequate background on the strengths
and weaknesses of CFD.
APPLICATION SPOTLIGHTS
Throughout the book are highlighted examples called Application Spotlights
where a real-world application of fluid mechanics is shown. A unique fea-
ture of these special examples is that they are written by guest authors. The
Application Spotlights are designed to show students how fluid mechanics
has diverse applications in a wide variety of fields. They also include eye-
catching photographs from the guest authors’ research.
CONVERSION FACTORS
Frequently used conversion factors, physical constants, and properties of air
and water at 20°C and atmospheric pressure are listed at the very end of the
book for easy reference.
NOMENCLATURE
A list of the major symbols, subscripts, and superscripts used in the text is
provided near the end of the book for easy reference.
xx
PREFACE

ACKNOWLEDGMENTS
The authors would like to acknowledge with appreciation the numerous and
valuable comments, suggestions, constructive criticisms, and praise from the
following evaluators and reviewers:
Bass Abushakra
Milwaukee School of Engineering
John G. Cherng
University of Michigan—Dearborn
Peter Fox
Arizona State University
Sathya Gangadbaran
Embry Riddle Aeronautical University
Jonathan Istok
Oregon State University
Tim Lee
McGill University
Nagy Nosseir
San Diego State University
Robert Spall
Utah State University
We also thank those who were acknowledged in the first, second, and
third editions of this book, but are too numerous to mention again here.
The authors are particularly grateful to Mehmet Kanoğlu of University of
Gaziantep for his valuable contributions, particularly his modifications of
end-of-chapter problems, his editing and updating of the solutions man-
ual, and his critical review of the entire manuscript. We also thank Tahsin
Engin of Sakarya University and Suat Canbazoğlu of Inonu University for
contributing several end-of-chapter problems, and Mohsen Hassan Vand
for reviewing the book and pointing out a number of errors.
Finally, special thanks must go to our families, especially our wives, Zehra
Çengel and Suzanne Cimbala, for their continued patience, understanding,
and support throughout the preparation of this book, which involved many
long hours when they had to handle family concerns on their own because
their husbands’ faces were glued to a computer screen.
Publishers are also thankful to the following faculty members for critically
reviewing the manuscripts:
Lajpat Rai
YMCA, Faridabad
Manoj Langhi
ITRAM, Gujarat
Masood Ahmed
MACET, Bihar
Yunus A. Çengel
John M. Cimbala
PREFACE
xxi

Online Resources for Instructors
Online Resources available at
http://www.mhhe.com/cengel/fm4
Your home page for teaching fluid mechanics, the Fluid Mechanics:
Fundamentals and Applications text-specific website is password protected
and offers resources for instructors.
■
■ Electronic Solutions Manual—provides PDF files with detailed typed

solutions to all text homework problems.
■
■ Lecture Slides—provide PowerPoint lecture slides for all chapters.

1
CHAPTER
1
INTRODUCTION AND
BASIC CONCEPTS
I
n this introductory chapter, we present the basic concepts commonly
used in the analysis of fluid flow. We start this chapter with a discussion
of the phases of matter and the numerous ways of classification of fluid
flow, such as viscous versus inviscid regions of flow, internal versus exter-
nal flow, compressible versus incompressible flow, laminar versus turbulent
flow, natural versus forced flow, and steady versus unsteady flow. We also
discuss the no-slip condition at solid–fluid interfaces and present a brief his-
tory of the development of fluid mechanics.
After presenting the concepts of system and control volume, we review
the unit systems that will be used. We then discuss how mathematical mod-
els for engineering problems are prepared and how to interpret the results
obtained from the analysis of such models. This is followed by a presenta-
tion of an intuitive systematic problem-solving technique that can be used as
a model in solving engineering problems. Finally, we discuss accuracy, pre-
cision, and significant digits in engineering measurements and calculations.
OBJECTIVES
When you finish reading this chapter,
you should be able to
■
■ Understand the basic concepts
of fluid mechanics
■
■ Recognize the various types
of fluid flow problems encoun-
tered in practice
■
■ Model engineering problems
and solve them in a systematic
manner
■
■ Have a working knowledge
of accuracy, precision,
and
significant digits, and

recognize the importance of
dimensional homogeneity in
engineering calculations
Schlieren image showing the thermal
plume produced by Professor Cimbala
as he welcomes you to the fascinating
world of ﬂuid mechanics.
Courtesy of Michael J. Hargather and John Cimbala.
FM_01.indd 1 4/15/2019 3:58:33 PM

2
Introduction and basic concepts
1–1 ■ INTRODUCTION
Mechanics is the oldest physical science that deals with both stationary and
moving bodies under the influence of forces. The branch of mechanics that
deals with bodies at rest is called statics, while the branch that deals with
bodies in motion under the action of forces is called dynamics. The subcat-
egory fluid mechanics is defined as the science that deals with the behavior
of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interac-
tion of fluids with solids or other fluids at the boundaries. Fluid mechanics
is also referred to as fluid dynamics by considering fluids at rest as a spe-
cial case of motion with zero velocity (Fig. 1–1).
Fluid mechanics itself is also divided into several categories. The study
of the motion of fluids that can be approximated as incompressible (such
as liquids, especially water, and gases at low speeds) is usually referred to
as hydrodynamics. A subcategory of hydrodynamics is hydraulics, which
deals with liquid flows in pipes and open channels. Gas dynamics deals
with the flow of fluids that undergo significant density changes, such as the
flow of gases through nozzles at high speeds. The category aerodynamics
deals with the flow of gases (especially air) over bodies such as aircraft,
rockets, and automobiles at high or low speeds. Some other specialized
categories such as meteorology, oceanography, and hydrology deal with
naturally occurring flows.
What Is a Fluid?
You will recall from physics that a substance exists in three primary phases:
solid, liquid, and gas. (At very high temperatures, it also exists as plasma.)
A substance in the liquid or gas phase is referred to as a fluid. Distinction
between a solid and a fluid is made on the basis of the substance’s abil-
ity to resist an applied shear (or tangential) stress that tends to change its
shape. A solid can resist an applied shear stress by deforming, whereas a
fluid deforms continuously under the influence of a shear stress, no matter
how small. In solids, stress is proportional to strain, but in fluids, stress is
proportional to strain rate. When a constant shear force is applied, a solid
eventually stops deforming at some fixed strain angle, whereas a fluid never
stops deforming and approaches a constant rate of strain.
Consider a rectangular rubber block tightly placed between two plates. As
the upper plate is pulled with a force F while the lower plate is held fixed,
the rubber block deforms, as shown in Fig. 1–2. The angle of deformation α
(called the shear strain or angular displacement) increases in proportion to
the applied force F. Assuming there is no slip between the rubber and the
plates, the upper surface of the rubber is displaced by an amount equal to
the displacement of the upper plate while the lower surface remains station-
ary. In equilibrium, the net force acting on the upper plate in the horizontal
direction must be zero, and thus a force equal and opposite to F must be
acting on the plate. This opposing force that develops at the plate–rubber
interface due to friction is expressed as F = τA, where τ is the shear stress
and A is the contact area between the upper plate and the rubber. When the
force is removed, the rubber returns to its original position. This phenome-
non would also be observed with other solids such as a steel block provided
that the applied force does not exceed the elastic range. If this experiment
were repeated with a fluid (with two large parallel plates placed in a large
Contact area,
A
Shear stress
τ = F/A
Shear
strain, α
Force, F
α
Deformed
rubber
FIGURE 1–2
Deformation of a rubber block placed
between two parallel plates under the
influence of a shear force. The shear
stress shown is that on the rubber—an
equal but opposite shear stress acts on
the upper plate.
FIGURE 1–1
Fluid mechanics deals with liquids and
gases in motion or at rest.
FM_01.indd 2 4/15/2019 3:58:36 PM

3
CHAPTER 1
body of water, for example), the fluid layer in contact with the upper plate
would move with the plate continuously at the velocity of the plate no mat-
ter how small the force F. The fluid velocity would decrease with depth
because of friction between fluid layers, reaching zero at the lower plate.
You will recall from statics that stress is defined as force per unit area
and is determined by dividing the force by the area upon which it acts. The
normal component of a force acting on a surface per unit area is called the
normal stress, and the tangential component of a force acting on a surface
per unit area is called shear stress (Fig. 1–3). In a fluid at rest, the normal
stress is called pressure. A fluid at rest is at a state of zero shear stress.
When the walls are removed or a liquid container is tilted, a shear develops
as the liquid moves to re-establish a horizontal free surface.
In a liquid, groups of molecules can move relative to each other, but the vol-
ume remains relatively constant because of the strong cohesive forces between
the molecules. As a result, a liquid takes the shape of the container it is in,
and it forms a free surface in a larger container in a gravitational field. A gas,
on the other hand, expands until it encounters the walls of the container and
fills the entire available space. This is because the gas molecules are widely
spaced, and the cohesive forces between them are very small. Unlike liquids,
a gas in an open container cannot form a free surface (Fig. 1–4).
Although solids and fluids are easily distinguished in most cases, this distinc-
tion is not so clear in some borderline cases. For example, asphalt appears and
behaves as a solid since it resists shear stress for short periods of time. When
these forces are exerted over extended periods of time, however, the asphalt
deforms slowly, behaving as a fluid. Some plastics, lead, and slurry mixtures
exhibit similar behavior. Such borderline cases are beyond the scope of this
text. The fluids we deal with in this text will be clearly recognizable as fluids.
Intermolecular bonds are strongest in solids and weakest in gases. One
reason is that molecules in solids are closely packed together, whereas in
gases they are separated by relatively large distances (Fig. 1–5). The mole-
cules in a solid are arranged in a pattern that is repeated throughout. Because
of the small distances between molecules in a solid, the attractive forces of
molecules on each other are large and keep the molecules at fixed positions.
The molecular spacing in the liquid phase is not much different from that of
Free surface
Liquid Gas
FIGURE 1–4
Unlike a liquid, a gas does not form a
free surface, and it expands to fill the
entire available space.
(a) (b) (c)
FIGURE 1–5
The arrangement of atoms in different phases: (a) molecules are at relatively fixed positions
in a solid, (b) groups of molecules move about each other in the liquid phase, and
(c) individual molecules move about at random in the gas phase.
FIGURE 1–3
The normal stress and shear stress at
the surface of a fluid element. For
fluids at rest, the shear stress is zero
and pressure is the only normal stress.
dFn
dFt
dF
Normal
to surface
Tangent
to surface
Force acting
on area dA
dA
Normal stress: 𝜎 =
dFn
dA
Shear stress: 𝜏 =
dFt
dA
FM_01.indd 3 4/15/2019 3:58:38 PM

4
the solid phase, except the molecules are no longer at fixed positions relative
to each other and they can rotate and translate freely. In a liquid, the inter-
molecular forces are weaker relative to solids, but still strong compared with
gases. The distances between molecules generally increase slightly as a solid
turns liquid, with water being a notable exception.
In the gas phase, the molecules are far apart from each other, and molecu-
lar ordering is nonexistent. Gas molecules move about at random, continu-
ally colliding with each other and the walls of the container in which they
are confined. Particularly at low densities, the intermolecular forces are very
small, and collisions are the only mode of interaction between the mole-
cules. Molecules in the gas phase are at a considerably higher energy level
than they are in the liquid or solid phase. Therefore, the gas must release a
large amount of its energy before it can condense or freeze.
Gas and vapor are often used as synonymous words. The vapor phase of
a substance is customarily called a gas when it is above the critical tempera-
ture. Vapor usually implies that the current phase is not far from a state of
condensation.
Any practical fluid system consists of a large number of molecules, and the
properties of the system naturally depend on the behavior of these molecules.
For example, the pressure of a gas in a container is the result of momentum
transfer between the molecules and the walls of the container. However, one
does not need to know the behavior of the gas molecules to determine the pres-
sure in the container. It is sufficient to attach a pressure gage to the container
(Fig. 1–6). This macroscopic or classical approach does not require a knowl-
edge of the behavior of individual molecules and provides a direct and easy
way to analyze engineering problems. The more elaborate microscopic or sta-
tistical approach, based on the average behavior of large groups of individual
molecules, is rather involved and is used in this text only in a supporting role.
Application Areas of Fluid Mechanics
It is important to develop a good understanding of the basic principles of
fluid mechanics, since fluid mechanics is widely used both in everyday
activities and in the design of modern engineering systems from vacuum
cleaners to supersonic aircraft. For example, fluid mechanics plays a vital
role in the human body. The heart is constantly pumping blood to all parts
of the human body through the arteries and veins, and the lungs are the sites
of airflow in alternating directions. All artificial hearts, breathing machines,
and dialysis systems are designed using fluid dynamics (Fig. 1–7).
An ordinary house is, in some respects, an exhibition hall filled with appli-
cations of fluid mechanics. The piping systems for water, natural gas, and
sewage for an individual house and the entire city are designed primarily on
the basis of fluid mechanics. The same is also true for the piping and ducting
network of heating and air-conditioning systems. A refrigerator involves tubes
through which the refrigerant flows, a compressor that pressurizes the refrig-
erant, and two heat exchangers where the refrigerant absorbs and rejects heat.
Fluid mechanics plays a major role in the design of all these components.
Even the operation of ordinary faucets is based on fluid mechanics.
We can also see numerous applications of fluid mechanics in an automo-
bile. All components associated with the transportation of the fuel from the
fuel tank to the cylinders—the fuel line, fuel pump, and fuel injectors or
Pressure
gage
FIGURE 1–6
On a microscopic scale, pressure
is determined by the interaction of
individual gas molecules. However,
we can measure the pressure on a

macroscopic scale with a pressure
gage.
FIGURE 1–7
Fluid dynamics is used extensively in
the design of artiﬁcial hearts. Shown
here is the Penn State Electric Total
Artiﬁcial Heart.
Courtesy of the Biomedical Photography Lab,
Penn State Biomedical Engineering Institute.
Used by permission.
FM_01.indd 4 4/15/2019 3:58:39 PM

5
CHAPTER 1
carburetors—as well as the mixing of the fuel and the air in the cylinders
and the purging of combustion gases in exhaust pipes—are analyzed using
fluid mechanics. Fluid mechanics is also used in the design of the heating
and air-conditioning system, the hydraulic brakes, the power steering, the
automatic transmission, the lubrication systems, the cooling system of the
engine block including the radiator and the water pump, and even the tires.
The sleek streamlined shape of recent model cars is the result of efforts to
minimize drag by using extensive analysis of flow over surfaces.
On a broader scale, fluid mechanics plays a major part in the design and
analysis of aircraft, boats, submarines, rockets, jet engines, wind turbines,
biomedical devices, cooling systems for electronic components, and trans-
portation systems for moving water, crude oil, and natural gas. It is also
considered in the design of buildings, bridges, and even billboards to make
sure that the structures can withstand wind loading. Numerous natural phe-
nomena such as the rain cycle, weather patterns, the rise of ground water to
the tops of trees, winds, ocean waves, and currents in large water bodies are
also governed by the principles of fluid mechanics (Fig. 1–8).
FIGURE 1–8
Some application areas of fluid mechanics.
Cars
© Ingram Publishing RF
Power plants
U.S. Nuclear Regulatory Commission (NRC)
Human body
© Jose Luis Pelaez Inc/Blend Images LLC RF
Piping and plumbing systems
Photo by John M. Cimbala
Wind turbines
© Mlenny Photography/Getty Images RF
Industrial applications
© 123RF
Aircraft and spacecraft
© Purestock/SuperStock/RF
Natural flows and weather
© Jochen Schlenker/Getty Images RF
Boats
© Doug Menuez/Getty Images RF
FM_01.indd 5 4/15/2019 3:58:42 PM

6
1–2 ■ A BRIEF HISTORY OF FLUID MECHANICS1
One of the first engineering problems humankind faced as cities were devel-
oped was the supply of water for domestic use and irrigation of crops. Our
urban lifestyles can be retained only with abundant water, and it is clear
from archeology that every successful civilization of prehistory invested in
the construction and maintenance of water systems. The Roman aqueducts,
some of which are still in use, are the best known examples. However, per-
haps the most impressive engineering from a technical viewpoint was done
at the Hellenistic city of Pergamon in present-day Turkey. There, from 283 to
133 bc, they built a series of pressurized lead and clay pipelines (Fig. 1–9),
up to 45 km long that operated at pressures exceeding 1.7 MPa (180 m of
head). Unfortunately, the names of almost all these early builders are lost to
history.
The earliest recognized contribution to fluid mechanics theory was made
by the Greek mathematician Archimedes (285–212 bc). He formulated and
applied the buoyancy principle in history’s first nondestructive test to deter-
mine the gold content of the crown of King Hiero II. The Romans built great
aqueducts and educated many conquered people on the benefits of clean
water, but overall had a poor understanding of fluids theory. (Perhaps they
shouldn’t have killed Archimedes when they sacked Syracuse.)
During the Middle Ages, the application of fluid machinery slowly but
steadily expanded. Elegant piston pumps were developed for dewatering
mines, and the watermill and windmill were perfected to grind grain, forge
metal, and for other tasks. For the first time in recorded human history, sig-
nificant work was being done without the power of a muscle supplied by a
person or animal, and these inventions are generally credited with enabling
the later industrial revolution. Again the creators of most of the progress
are unknown, but the devices themselves were well documented by several
technical writers such as Georgius Agricola (Fig. 1–10).
The Renaissance brought continued development of fluid systems and
machines, but more importantly, the scientific method was perfected and
adopted throughout Europe. Simon Stevin (1548–1617), Galileo Galilei
(1564–1642), Edme Mariotte (1620–1684), and Evangelista Torricelli
(1608–1647) were among the first to apply the method to fluids as they
investigated hydrostatic pressure distributions and vacuums. That work
was integrated and refined by the brilliant mathematician and philosopher,
Blaise Pascal (1623–1662). The Italian monk, Benedetto Castelli (1577–
1644) was the first person to publish a statement of the continuity principle
for fluids. Besides formulating his equations of motion for solids, Sir Isaac
Newton (1643–1727) applied his laws to fluids and explored fluid inertia
and resistance, free jets, and viscosity. That effort was built upon by Daniel
Bernoulli (1700–1782), a Swiss, and his associate Leonard Euler (1707–
1783). Together, their work defined the energy and momentum equations.
Bernoulli’s 1738 classic treatise Hydrodynamica may be considered the first
fluid mechanics text. Finally, Jean d’Alembert (1717–1789) developed the
idea of velocity and acceleration components, a differential expression of
1
This section is contributed by Professor Glenn Brown of Oklahoma State University.
FIGURE 1–9
Segment of Pergamon pipeline.
Each clay pipe section was
13 to 18 cm in diameter.
germesa/Shutterstock.
FIGURE 1–10
A mine hoist powered
by a reversible water wheel.
© Universal History Archive/Getty Images
FM_01.indd 6 4/10/21 7:24 PM

7
CHAPTER 1
continuity, and his “paradox” of zero resistance to steady uniform motion
over a body.
The development of fluid mechanics theory through the end of the eigh-
teenth century had little impact on engineering since fluid properties and
parameters were poorly quantified, and most theories were abstractions that
could not be quantified for design purposes. That was to change with the
development of the French school of engineering led by Riche de Prony
(1755–1839). Prony (still known for his brake to measure shaft power) and
his associates in Paris at the École Polytechnique and the École des Ponts
et Chaussées were the first to integrate calculus and scientific theory into
the engineering curriculum, which became the model for the rest of the
world. (So now you know whom to blame for your painful freshman year.)
Antonie Chezy (1718–1798), Louis Navier (1785–1836), Gaspard Coriolis
(1792–1843), Henry Darcy (1803–1858), and many other contributors to
fluid engineering and theory were students and/or instructors at the schools.
By the mid nineteenth century, fundamental advances were coming on
several fronts. The physician Jean Poiseuille (1799–1869) had accurately
measured flow in capillary tubes for multiple fluids, while in Germany
Gotthilf Hagen (1797–1884) had differentiated between laminar and turbu-
lent flow in pipes. In England, Lord Osborne Reynolds (1842–1912) con-
tinued that work (Fig. 1–11) and developed the dimensionless number that
bears his name. Similarly, in parallel to the early work of Navier, George
Stokes (1819–1903) completed the general equation of fluid motion (with
friction) that takes their names. William Froude (1810–1879) almost single-
handedly developed the procedures and proved the value of physical model
testing. American expertise had become equal to the Europeans as demon-
strated by James Francis’ (1815–1892) and Lester Pelton’s (1829–1908) pio-
neering work in turbines and Clemens Herschel’s (1842–1930) invention of
the Venturi meter.
In addition to Reynolds and Stokes, many notable contributions were made
to fluid theory in the late nineteenth century by Irish and English
scientists,
including William Thomson, Lord Kelvin (1824–1907), William Strutt, Lord
Rayleigh (1842–1919), and Sir Horace Lamb (1849–1934). These individu-
als investigated a large number of problems, including dimensional analysis,
irrotational flow, vortex motion, cavitation, and waves. In a broader sense,
FIGURE 1–11
Osborne Reynolds’ original apparatus
for demonstrating the onset of turbu-
lence in pipes, being operated
by John Lienhard at the University
of Manchester in 1975.
Courtesy of John Lienhard, University of Houston.
Used by Permission.
FM_01.indd 7 4/15/2019 3:58:45 PM

8
their work also explored the links between fluid mechanics, thermodynam-
ics, and heat transfer.
The dawn of the twentieth century brought two monumental developments.
First, in 1903, the self-taught Wright brothers (Wilbur, 1867–1912; Orville,
1871–1948) invented the airplane through application of theory and deter-
mined experimentation. Their primitive invention was complete and contained
all the major aspects of modern aircraft (Fig. 1–12). The Navier–Stokes equa-
tions were of little use up to this time because they were too difficult to solve.
In a pioneering paper in 1904, the German Ludwig Prandtl (1875–1953)
showed that fluid flows can be divided into a layer near the walls, the bound-
ary layer, where the friction effects are significant, and an outer layer where
such effects are negligible and the simplified Euler and Bernoulli equations
are applicable. His students, Theodor von Kármán (1881–1963), Paul Blasius
(1883–1970), Johann Nikuradse (1894–1979), and others, built on that theory
in both hydraulic and aerodynamic applications. (During World War II, both
sides benefited from the theory as Prandtl remained in Germany while his
best student, the Hungarian-born von Kármán, worked in America.)
The mid twentieth century could be considered a golden age of fluid
mechanics applications. Existing theories were adequate for the tasks at
hand, and fluid properties and parameters were well defined. These sup-
ported a huge expansion of the aeronautical, chemical, industrial, and
water resources sectors; each of which pushed fluid mechanics in new
directions. Fluid mechanics research and work in the late twentieth century
were dominated by the development of the digital computer in America.
The ability to solve large complex problems, such as global climate mod-
eling or the optimization of a turbine blade, has provided a benefit to our
society that the eighteenth-century developers of fluid mechanics could
never have imagined (Fig. 1–13). The principles presented in the following
pages have been applied to flows ranging from a moment at the micro-
scopic scale to 50 years of simulation for an entire river basin. It is truly
mind-boggling.
Where will fluid mechanics go in the twenty-first century and beyond?
Frankly, even a limited extrapolation beyond the present would be sheer folly.
However, if history tells us anything, it is that engineers will be applying
what they know to benefit society, researching what they don’t know, and
having a great time in the process.
1–3 ■ THE NO-SLIP CONDITION
Fluid flow is often confined by solid surfaces, and it is important to under-
stand how the presence of solid surfaces affects fluid flow. We know that
water in a river cannot flow through large rocks, and must go around them.
That is, the water velocity normal to the rock surface must be zero, and
water approaching the surface normally comes to a complete stop at the sur-
face. What is not as obvious is that water approaching the rock at any angle
also comes to a complete stop at the rock surface, and thus the tangential
velocity of water at the surface is also zero.
Consider the flow of a fluid in a stationary pipe or over a solid surface
that is nonporous (i.e., impermeable to the fluid). All experimental observa-
tions indicate that a fluid in motion comes to a complete stop at the surface
FIGURE 1–12
The Wright brothers take
flight at Kitty Hawk.
Courtesy Library of Congress Prints
Photographs Division [LC-DIG-ppprs-00626].
FIGURE 1–13
Old and new wind turbine technologies
north of Woodward, OK. The modern
turbines have up to 8 MW capacities.
oorka/Shutterstock.
FM_01.indd 8 4/10/21 9:28 PM

9
CHAPTER 1
and assumes a zero velocity relative to the surface. That is, a fluid in direct
contact with a solid “sticks” to the surface, and there is no slip. This is
known as the no-slip condition. The fluid property responsible for the no-
slip condition and the development of the boundary layer is viscosity and is
discussed in Chap. 2.
The photograph in Fig. 1–14 clearly shows the evolution of a velocity gra-
dient as a result of the fluid sticking to the surface of a blunt nose. The layer
that sticks to the surface slows the adjacent fluid layer because of viscous
forces between the fluid layers, which slows the next layer, and so on. A
consequence of the no-slip condition is that all velocity profiles must have
zero values with respect to the surface at the points of contact between a
fluid and a solid surface (Fig. 1–15). Therefore, the no-slip condition is
responsible for the development of the velocity profile. The flow region
adjacent to the wall in which the viscous effects (and thus the velocity gra-
dients) are significant is called the boundary layer. Another consequence
of the no-slip condition is the surface drag, or skin friction drag, which is
the force a fluid exerts on a surface in the flow direction.
When a fluid is forced to flow over a curved surface, such as the back
side of a cylinder, the boundary layer may no longer remain attached to the
sur
face and separates from the surface—a process called flow separation
(Fig. 1–16). We emphasize that the no-slip condition applies everywhere
along the surface, even downstream of the separation point. Flow separation
is discussed in greater detail in Chap. 9.
A phenomenon similar to the no-slip condition occurs in heat transfer.
When two bodies at different temperatures are brought into contact, heat
transfer occurs such that both bodies assume the same temperature at the
points of contact. Therefore, a fluid and a solid surface have the same tem-
perature at the points of contact. This is known as no-temperature-jump
condition.
1–4 ■ CLASSIFICATION OF FLUID FLOWS
Earlier we defined fluid mechanics as the science that deals with the behav-
ior of fluids at rest or in motion, and the interaction of fluids with solids or
other fluids at the boundaries. There is a wide variety of fluid flow prob-
lems encountered in practice, and it is usually convenient to classify them
on the basis of some common characteristics to make it feasible to study
them in groups. There are many ways to classify fluid flow problems, and
here we present some general categories.
FIGURE 1–14
The development of a velocity profile
due to the no-slip condition as a fluid
flows over a blunt nose.
“Hunter Rouse: Laminar and Turbulence Flow
Film.” Copyright IIHR-Hydroscience Engineer-
ing. The University of Iowa. Used by permission.
Relative
velocities
of fluid layers
Uniform
approach
velocity, V
Zero
velocity
at the
surface
Plate
FIGURE 1–15
A fluid flowing over a stationary
surface comes to a complete stop at
the surface because of the no-slip
condition.
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10
Viscous versus Inviscid Regions of Flow
When two fluid layers move relative to each other, a friction force devel-
ops between them and the slower layer tries to slow down the faster layer.
This internal resistance to flow is quantified by the fluid property viscosity,
which is a measure of internal stickiness of the fluid. Viscosity is caused
by cohesive forces between the molecules in liquids and by molecular colli-
sions in gases. There is no fluid with zero viscosity, and thus all fluid flows
involve viscous effects to some degree. Flows in which the frictional effects
are significant are called viscous flows. However, in many flows of practi-
cal interest, there are regions (typically regions not close to solid surfaces)
where viscous forces are negligibly small compared to inertial or pressure
forces. Neglecting the viscous terms in such inviscid flow regions greatly
simplifies the analysis without much loss in accuracy.
The development of viscous and inviscid regions of flow as a result of
inserting a flat plate parallel into a fluid stream of uniform velocity is shown
in Fig. 1–17. The fluid sticks to the plate on both sides because of the no-slip
condition, and the thin boundary layer in which the viscous effects are signifi-
cant near the plate surface is the viscous flow region. The region of flow on
both sides away from the plate and largely unaffected by the presence of the
plate is the inviscid flow region.
Internal versus External Flow
A fluid flow is classified as being internal or external, depending on whether
the fluid flows in a confined space or over a surface. The flow of an
unbounded fluid over a surface such as a plate, a wire, or a pipe is
external
flow. The flow in a pipe or duct is internal flow if the fluid is bounded
by solid surfaces. Water flow in a pipe, for example, is internal flow, and

airflow over a ball or over an exposed pipe during a windy day is external
flow (Fig. 1–18). The flow of liquids in a duct is called open-channel flow if
the duct is only partially filled with the liquid and there is a free surface. The
flows of water in rivers and irrigation ditches are examples of such flows.
Internal flows are dominated by the influence of viscosity throughout
the flow field. In external flows the viscous effects are limited to boundary

layers near solid surfaces and to wake regions downstream of bodies.
Compressible versus Incompressible Flow
A flow is classified as being compressible or incompressible, depending
on the level of variation of density during flow. Incompressibility is an
approximation, in which the flow is said to be incompressible if the density
remains nearly constant throughout. Therefore, the volume of every portion
of fluid remains unchanged over the course of its motion when the flow is
approximated as incompressible.
The densities of liquids are essentially constant, and thus the flow of liq-
uids is typically incompressible. Therefore, liquids are usually referred to as
incompressible substances. A pressure of 210 atm, for example, causes the
density of liquid water at 1 atm to change by just 1 percent. Gases, on the
other hand, are highly compressible. A pressure change of just 0.01 atm, for
example, causes a change of 1 percent in the density of atmospheric air.
FIGURE 1–18
External flow over a tennis ball, and
the turbulent wake region behind.
Courtesy of NASA and Cislunar Aerospace, Inc.
Inviscid flow
region
Viscous flow
region
Inviscid flow
region
FIGURE 1–17
The flow of an originally uniform
fluid stream over a flat plate, and
the regions of viscous flow (next to
the plate on both sides) and inviscid
flow (away from the plate).
Fundamentals of Boundry Layers,
National Committee for Fluid Mechanics Films,
© Education Development Center.
FM_01.indd 10 4/15/2019 3:58:47 PM

11
CHAPTER 1
When analyzing rockets, spacecraft, and other systems that involve high-
speed gas flows (Fig. 1–19), the flow speed is often expressed in terms of
the dimensionless Mach number defined as
Ma =
V
c
=
Speed of flow
Speed of sound
where c is the speed of sound whose value is 346 m/s in air at room tempera-
ture at sea level. A flow is called sonic when Ma = 1, subsonic when Ma 1,
supersonic when Ma 1, and hypersonic when Ma ≫ 1. Dimensionless
parameters are discussed in detail in Chap. 7. Compressible flow is discussed
in detail in Chap. 12.
Liquid flows are incompressible to a high level of accuracy, but the level
of variation of density in gas flows and the consequent level of approxi-
mation made when modeling gas flows as incompressible depends on the
Mach number. Gas flows can often be approximated as incompressible if the
density changes are under about 5 percent, which is usually the case when
Ma 0.3. Therefore, the compressibility effects of air at room tempera-
ture can be neglected at speeds under about 100 m/s. Compressibility effects
should never be neglected for supersonic flows, however, since compress-
ible flow phenomena like shock waves occur (Fig. 1–19).
Small density changes of liquids corresponding to large pressure changes
can still have important consequences. The irritating “water hammer” in a
water pipe, for example, is caused by the vibrations of the pipe generated by
the reflection of pressure waves following the sudden closing of the valves.
Laminar versus Turbulent Flow
Some flows are smooth and orderly while others are rather chaotic. The
highly ordered fluid motion characterized by smooth layers of fluid is called
laminar. The word laminar comes from the movement of adjacent fluid
particles together in “laminae.” The flow of high-viscosity fluids such as
oils at low velocities is typically laminar. The highly disordered fluid motion
that typically occurs at high velocities and is characterized by velocity fluc-
tuations is called turbulent (Fig. 1–20). The flow of low-viscosity fluids
such as air at high velocities is typically turbulent. A flow that alternates
between being laminar and turbulent is called transitional. The experiments
conducted by Osborne Reynolds in the 1880s resulted in the establishment
of the dimensionless Reynolds number, Re, as the key parameter for the
determination of the flow regime in pipes (Chap. 8).
Natural (or Unforced) versus Forced Flow
A fluid flow is said to be natural or forced, depending on how the fluid
motion is initiated. In forced flow, a fluid is forced to flow over a surface
or in a pipe by external means such as a pump or a fan. In natural flows,
fluid motion is due to natural means such as the buoyancy effect, which
manifests itself as the rise of warmer (and thus lighter) fluid and the fall of
cooler (and thus denser) fluid (Fig. 1–21). In solar hot-water systems, for
example, the thermosiphoning effect is commonly used to replace pumps by
placing the water tank sufficiently above the solar collectors.
Laminar
Transitional
Turbulent
FIGURE 1–20
Laminar, transitional, and turbulent
flows over a flat plate.
Courtesy of ONERA. Photo by Werlé.
FIGURE 1–19
Schlieren image of the spherical shock
wave produced by a bursting ballon
at the Penn State Gas Dynamics Lab.
Several secondary shocks are seen in
the air surrounding the ballon.
© G.S. Settles, Gas Dynamics Lab, Penn State
University. Used with permission.
FM_01.indd 11 4/15/2019 3:58:48 PM

12
Steady versus Unsteady Flow
The terms steady and uniform are used frequently in engineering, and thus
it is important to have a clear understanding of their meanings. The term
steady implies no change of properties, velocity, temperature, etc., at a point
with time. The opposite of steady is unsteady. The term uniform implies no
change with location over a specified region. These meanings are consistent
with their everyday use (steady girlfriend, uniform distribution, etc.).
The terms unsteady and transient are often used interchangeably, but these
terms are not synonyms. In fluid mechanics, unsteady is the most general term
that applies to any flow that is not steady, but transient is typically used for
developing flows. When a rocket engine is fired up, for example, there are tran-
sient effects (the pressure builds up inside the rocket engine, the flow accelerates,
etc.) until the engine settles down and operates steadily. The term periodic refers
to the kind of unsteady flow in which the flow oscillates about a steady mean.
Many devices such as turbines, compressors, boilers, condensers, and heat
exchangers operate for long periods of time under the same conditions, and they
are classified as steady-flow devices. (Note that the flow field near the rotating
blades of a turbomachine is of course unsteady, but we consider the overall
flow field rather than the details at some localities when we classify devices.)
During steady flow, the fluid properties can change from point to point within
a device, but at any fixed point they remain constant. Therefore, the volume,
the mass, and the total energy content of a steady-flow device or flow section
remain constant in steady operation. A simple analogy is shown in Fig. 1–22.
Steady-flow conditions can be closely approximated by devices that are
intended for continuous operation such as turbines, pumps, boilers, condens-
ers, and heat exchangers of power plants or refrigeration systems. Some
cyclic devices, such as reciprocating engines or compressors, do not sat-
isfy the steady-flow conditions since the flow at the inlets and the exits is
pulsating and not steady. However, the fluid properties vary with time in a
FIGURE 1–21
In this schlieren image of a girl in a
swimming suit, the rise of lighter,
warmer air adjacent to her body
indicates that humans and warm-
blooded animals are surrounded by
thermal plumes of rising warm air.
© G.S. Settles, Gas Dynamics Lab,
Penn State University. Used with permission.
FIGURE 1–22
Comparison of (a) instantaneous
snapshot of an unsteady flow, and
(b) long exposure picture of the
same flow.
Mongkol Saikhunthod/Shutterstock, AlessandraRC/
Shutterstock. (a) (b)
FM_01.indd 12 4/10/21 9:37 PM

13
CHAPTER 1
periodic manner, and the flow through these devices can still be analyzed as
a steady-flow process by using time-averaged values for the properties.
Some fascinating visualizations of fluid flow are provided in the book An
Album of Fluid Motion by Milton Van Dyke (1982). A nice illustration of
an unsteady-flow field is shown in Fig. 1–23, taken from Van Dyke’s book.
Figure 1–23a is an instantaneous snapshot from a high-speed motion picture; it
reveals large, alternating, swirling, turbulent eddies that are shed into the periodi-
cally oscillating wake from the blunt base of the object. The unsteady wake pro-
duces waves that move upstream alternately over the top and bottom surfaces of
the airfoil in an unsteady fashion. Figure 1–23b shows the same flow field, but
the film is exposed for a longer time so that the image is time averaged over 12
cycles. The resulting time-averaged flow field appears “steady” since the details
of the unsteady oscillations have been lost in the long exposure.
One of the most important jobs of an engineer is to determine whether it is
sufficient to study only the time-averaged “steady” flow features of a problem,
or whether a more detailed study of the unsteady features is required. If the engi-
neer were interested only in the overall properties of the flow field (such as the
time-averaged drag coefficient, the mean velocity, and pressure fields), a time-
averaged description like that of Fig. 1–23b, time-averaged experimental mea-
surements, or an analytical or numerical calculation of the time-averaged flow
field would be sufficient. However, if the engineer were interested in details
about the unsteady-flow field, such as flow-induced vibrations, unsteady pres-
sure fluctuations, or the sound waves emitted from the turbulent eddies or the
shock waves, a time-averaged description of the flow field would be insufficient.
Most of the analytical and computational examples provided in this text-
book deal with steady or time-averaged flows, although we occasionally
point out some relevant unsteady-flow features as well when appropriate.
One-, Two-, and Three-Dimensional Flows
A flow field is best characterized by its velocity distribution, and thus a flow
is said to be one-, two-, or three-dimensional if the flow velocity varies in
one, two, or three primary dimensions, respectively. A typical fluid flow
involves a three-dimensional geometry, and the velocity may vary in all three
dimensions, rendering the flow three-dimensional [V
›
(x, y, z) in rectangular
or V
›
(r, θ, z) in cylindrical coordinates]. However, the variation of velocity in
certain directions can be small relative to the variation in other directions and
can be ignored with negligible error. In such cases, the flow can be modeled
conveniently as being one- or two-dimensional, which is easier to analyze.
Consider steady flow of a fluid entering from a large tank into a circular
pipe. The fluid velocity everywhere on the pipe surface is zero because
of the no-slip condition, and the flow is two-dimensional in the entrance
region of the pipe since the velocity changes in both the r- and z-directions,
but not in the θ-direction. The velocity profile develops fully and remains
unchanged after some distance from the inlet (about 10 pipe diameters in
turbulent flow, and typically farther than that in laminar pipe flow, as in
Fig. 1–24), and the flow in this region is said to be fully developed. The
fully developed flow in a circular pipe is one-dimensional since the velocity
varies in the radial r-direction but not in the angular θ- or axial z-directions,
as shown in Fig. 1–24. That is, the velocity profile is the same at any axial
z-location, and it is symmetric about the axis of the pipe.
(a)
(b)
FIGURE 1–23
Oscillating wake of a blunt-based
airfoil at Mach number 0.6. Photo (a)
is an instantaneous image, while
photo (b) is a long-exposure
(time-averaged) image.
(a) Dyment, A., Flodrops, J. P. Gryson, P.
1982 in Flow Visualization II, W. Merzkirch, ed.,
331–336. Washington: Hemisphere. Used by

permission of Arthur Dyment.
(b) Dyment, A. Gryson, P. 1978 in Inst. Mèc.
Fluides Lille, No. 78-5. Used by permission of
Arthur Dyment.
FM_01.indd 13 4/15/2019 3:58:51 PM

14
Note that the dimensionality of the flow also depends on the choice of
coordinate system and its orientation. The pipe flow discussed, for example, is
one-dimensional in cylindrical coordinates, but two-dimensional in Cartesian
coordinates—illustrating the importance of choosing the most appropriate
coordinate system. Also note that even in this simple flow, the velocity cannot
be uniform across the cross section of the pipe because of the no-slip condi-
tion. However, at a well-rounded entrance to the pipe, the velocity profile may
be approximated as being nearly uniform across the pipe, since the velocity is
nearly constant at all radii except very close to the pipe wall.
A flow may be approximated as two-dimensional when the aspect ratio is
large and the flow does not change appreciably along the longer dimension. For
example, the flow of air over a car antenna can be considered two-dimensional
except near its ends since the antenna’s length is much greater than its diam-
eter, and the airflow hitting the antenna is fairly uniform (Fig. 1–25).
FIGURE 1–25
Flow over a car antenna is
approximately two-dimensional
except near the top and bottom
of the antenna.
Axis of
symmetry
r
z
θ
FIGURE 1–26
Axisymmetric flow over a bullet.
z
r
Developing velocity
profile, V(r, z)
Fully developed
velocity profile, V(r)
FIGURE 1–24
The development of the velocity
profile in a circular pipe. V = V(r, z)
and thus the flow is two-dimensional
in the entrance region, and becomes
one-dimensional downstream when
the velocity profile fully develops
and remains unchanged in the flow
direction, V = V(r).
EXAMPLE 1–1 Axisymmetric Flow over a Bullet
Consider a bullet piercing through calm air during a short time interval in which the
bullet’s speed is nearly constant. Determine if the time-averaged airflow over the bullet
during its flight is one-, two-, or three-dimensional (Fig. 1–26).
SOLUTION It is to be determined whether airflow over a bullet is one-, two-, or
three-dimensional.
Assumptions There are no significant winds and the bullet is not spinning.
Analysis The bullet possesses an axis of symmetry and is therefore an axisym-
metric body. The airflow upstream of the bullet is parallel to this axis, and we
expect the time-averaged airflow to be rotationally symmetric about the axis—such
flows are said to be axisymmetric. The velocity in this case varies with axial dis-
tance z and radial distance r, but not with angle θ. Therefore, the time-averaged
airflow over the bullet is two-dimensional.
Discussion While the time-averaged airflow is axisymmetric, the instantaneous
airflow is not, as illustrated in Fig. 1–23. In Cartesian coordinates, the flow would
be three-dimensional. Finally, many bullets also spin.
Uniform versus Nonuniform Flow
Uniform flow implies that all fluid properties, such as velocity, pressure, tem-
perature, etc., do not vary with position. A wind tunnel test section, for exam-
ple, is designed such that the air flow is as uniform as possible. Even then,
however, the flow does not remain uniform as we approach the wind tun-
nel walls, due to the no-slip condition and the presence of a boundary layer,
FM_01.indd 14 4/15/2019 3:58:52 PM

15
CHAPTER 1
as mentioned previously. The flow just downstream of a well-rounded pipe
entrance (Fig. 1–24) is nearly uniform, again except for a very thin bound-
ary layer near the wall. In engineering practice, it is common to approximate
the flow in ducts and pipes and at inlets and outlets as uniform, even when
it is not, for simplicity in calculations. For example, the fully developed pipe
flow velocity profile of Fig. 1–24 is certainly not uniform, but for calculation
purposes we sometimes approximate it as the uniform profile at the far left
of the pipe, which has the same average velocity. Although this makes the
calculations easier, it also introduces some errors that require correction fac-
tors; these are discussed in Chaps. 5 and 6 for kinetic energy and momentum,
respectively.
1–5 ■ SYSTEM AND CONTROL VOLUME
A system is defined as a quantity of matter or a region in space chosen for
study. The mass or region outside the system is called the surroundings.
The real or imaginary surface that separates the system from its surround-
ings is called the boundary (Fig. 1–27). The boundary of a system can be
fixed or movable. Note that the boundary is the contact surface shared by
both the system and the surroundings. Mathematically speaking, the bound-
ary has zero thickness, and thus it can neither contain any mass nor occupy
any volume in space.
Systems may be considered to be closed or open, depending on whether
a fixed mass or a volume in space is chosen for study. A closed system
(also known as a control mass or simply a system when the context makes
it clear) consists of a fixed amount of mass, and no mass can cross its
boundary. But energy, in the form of heat or work, can cross the boundary,
and the volume of a closed system does not have to be fixed. If, as a special
case, even energy is not allowed to cross the boundary, that system is called
an isolated system.
Consider the piston–cylinder device shown in Fig. 1–28. Let us say that
we would like to find out what happens to the enclosed gas when it is
heated. Since we are focusing our attention on the gas, it is our system. The
inner surfaces of the piston and the cylinder form the boundary, and since
no mass is crossing this boundary, it is a closed system. Notice that energy
may cross the boundary, and part of the boundary (the inner surface of the
piston, in this case) may move. Everything outside the gas, including the
piston and the cylinder, is the surroundings.
An open system, or a control volume, as it is often called, is a selected
region in space. It usually encloses a device that involves mass flow such as
a compressor, turbine, or nozzle. Flow through these devices is best stud-
ied by selecting the region within the device as the control volume. Both
mass and energy can cross the boundary (the control surface) of a control
volume.
A large number of engineering problems involve mass flow in and out
of an open system and, therefore, are modeled as control volumes. A water
heater, a car radiator, a turbine, and a compressor all involve mass flow
and should be analyzed as control volumes (open systems) instead of as
control masses (closed systems). In general, any arbitrary region in space
can be selected as a control volume. There are no concrete rules for the
SURROUNDINGS
BOUNDARY
SYSTEM
FIGURE 1–27
System, surroundings, and boundary.
GAS
2 kg
1.5 m3
GAS
2 kg
1 m3
Moving
boundary
Fixed
boundary
FIGURE 1–28
A closed system with a moving
boundary.
FM_01.indd 15 4/15/2019 3:58:52 PM

16
selection of control volumes, but a wise choice certainly makes the analysis
much easier. If we were to analyze the flow of air through a nozzle, for
example, a good choice for the control volume would be the region within
the nozzle, or perhaps surrounding the entire nozzle.
A control volume can be fixed in size and shape, as in the case of a noz-
zle, or it may involve a moving boundary, as shown in Fig. 1–29. Most con-
trol volumes, however, have fixed boundaries and thus do not involve any
moving boundaries. A control volume may also involve heat and work inter-
actions just as a closed system, in addition to mass interaction.
1–6 ■ IMPORTANCE OF DIMENSIONS
AND UNITS
Any physical quantity can be characterized by dimensions. The magnitudes
assigned to the dimensions are called units. Some basic dimensions such
as mass m, length L, time t, and temperature T are selected as primary or
fundamental dimensions, while others such as velocity V, energy E, and
volume V are expressed in terms of the primary dimensions and are called
secondary dimensions, or derived dimensions.
A number of unit systems have been developed over the years. Despite
strong efforts in the scientific and engineering community to unify the
world with a single unit system, two sets of units are still in common use
today: the English system, which is also known as the United States Cus-
tomary System (USCS), and the metric SI (from Le Système International
d’ Unités), which is also known as the International System. The SI is a
simple and logical system based on a decimal relationship between the vari-
ous units, and it is being used for scientific and engineering work in most of
the industrialized nations, including England. The English system, however,
has no apparent systematic numerical base, and various units in this system
are related to each other rather arbitrarily (12 in = 1 ft, 1 mile = 5280 ft,
4 qt = 1 gal, etc.), which makes it confusing and difficult to learn. The
United States is the only industrialized country that has not yet fully con-
verted to the metric system.
The systematic efforts to develop a universally acceptable system of
units dates back to 1790 when the French National Assembly charged the
French Academy of Sciences to come up with such a unit system. An early
version of the metric system was soon developed in France, but it did not
find universal acceptance until 1875 when The Metric Convention Treaty
was prepared and signed by 17 nations, including the United States. In this
international treaty, meter and gram were established as the metric units
for length and mass, respectively, and a General Conference of Weights
and Measures (CGPM) was established that was to meet every six years.
In 1960, the CGPM produced the SI, which was based on six fundamental
quantities, and their units were adopted in 1954 at the Tenth General Con-
ference of Weights and Measures: meter (m) for length, kilogram (kg) for
mass, second (s) for time, ampere (A) for electric current, degree Kelvin (°K)
for temperature, and candela (cd) for luminous intensity (amount of light).
In 1971, the CGPM added a seventh fundamental quantity and unit: mole
(mol) for the amount of matter.
FIGURE 1–29
A control volume may involve
fixed, moving, real, and imaginary
boundaries.
CV
Moving
boundary
Fixed
boundary
Real boundary
(b) A control volume (CV) with fixed and
moving boundaries as well as real and
imaginary boundaries
(a) A control volume (CV) with real and
imaginary boundaries
Imaginary
boundary
CV
(a nozzle)
FM_01.indd 16 4/15/2019 3:58:53 PM

17
CHAPTER 1
Based on the notational scheme introduced in 1967, the degree sym-
bol was officially dropped from the absolute temperature unit, and all
unit names were to be written without capitalization even if they were
derived from proper names (Table 1–1). However, the abbreviation of a
unit was to be capitalized if the unit was derived from a proper name.
For example, the SI unit of force, which is named after Sir Isaac Newton
(1647–1723), is newton (not Newton), and it is abbreviated as N. Also,
the full name of a unit may be pluralized, but its abbreviation cannot. For
example, the length of an object can be 5 m or 5 meters, not 5 ms or 5
meter. Finally, no period is to be used in unit abbreviations unless they
appear at the end of a sentence. For example, the proper abbreviation of
meter is m (not m.).
The recent move toward the metric system in the United States seems to
have started in 1968 when Congress, in response to what was happening
in the rest of the world, passed a Metric Study Act. Congress continued
to promote a voluntary switch to the metric system by passing the Metric
Conversion Act in 1975. A trade bill passed by Congress in 1988 set a
September 1992 deadline for all federal agencies to convert to the metric
system. However, the deadlines were relaxed later with no clear plans for
the future.
As pointed out, the SI is based on a decimal relationship between units. The
prefixes used to express the multiples of the various units are listed in Table 1–2.
They are standard for all units, and the student is encouraged to memorize some
of them because of their widespread use (Fig. 1–30).
Some SI and English Units
In SI, the units of mass, length, and time are the kilogram (kg), meter (m),
and second (s), respectively. The respective units in the English system are
the pound-mass (lbm), foot (ft), and second (s). The pound symbol lb is
actually the abbreviation of libra, which was the ancient Roman unit of
weight. The English retained this symbol even after the end of the Roman
occupation of Britain in 410. The mass and length units in the two systems
are related to each other by
1 lbm = 0.45359 kg
1 ft = 0.3048 m
In the English system, force is often considered to be one of the primary
dimensions and is assigned a nonderived unit. This is a source of confu-
sion and error that necessitates the use of a dimensional constant (gc) in
many formulas. To avoid this nuisance, we consider force to be a secondary
dimension whose unit is derived from Newton’s second law, i.e.,
Force = (Mass) (Acceleration)
or F = ma (1–1)
In SI, the force unit is the newton (N), and it is defined as the force required
to accelerate a mass of 1 kg at a rate of 1 m/s2
. In the English system, the
force unit is the pound-force (lbf) and is defined as the force required to
TABLE 1–1
The seven fundamental (or primary)
dimensions and their units in SI
Dimension Unit
Length
Mass
Time
Temperature
Electric current
Amount of light
Amount of matter
meter (m)
kilogram (kg)
second (s)
kelvin (K)
ampere (A)
candela (cd)
mole (mol)
TABLE 1–2
Standard prefixes in SI units
Multiple Prefix
1024
1021
1018
1015
1012
109
106
103
102
101
10−1
10−2
10−3
10−6
10−9
10−12
10−15
10−18
10−21
10−24
yotta, Y
zetta, Z
exa, E
peta, P
tera, T
giga, G
mega, M
kilo, k
hecto, h
deka, da
deci, d
centi, c
milli, m
micro, μ
nano, n
pico, p
femto, f
atto, a
zepto, z
yocto, y
1 kg
200 mL
(0.2 L) (103 g)
1 MΩ
(106
Ω)
FIGURE 1–30
The SI unit prefixes are used in all
branches of engineering.
FM_01.indd 17 4/15/2019 3:58:54 PM

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